Domination number of an interval catch digraph family and its use for testing uniformity

Abstract

We consider a special type of interval catch digraph (ICD) family for one-dimensional data in a randomized setting and propose its use for testing uniformity. These ICDs are defined with an expansion and a centrality parameter, hence we will refer to this ICD as parameterized ICD (PICD). We derive the exact (and asymptotic) distribution of the domination number of this PICD family when its vertices are from a uniform (and non-uniform) distribution in one dimension for the entire range of the parameters; thereby determine the parameters for which the asymptotic distribution is non-degenerate. We observe jumps (from degeneracy to non-degeneracy or from a non-degenerate distribution to another) in the asymptotic distribution of the domination number at certain parameter combinations. We use the domination number for testing uniformity of data in real line, prove its consistency against certain alternatives, and compare it with two commonly used tests and three recently proposed tests in literature and also arc density of this ICD and of another ICD family in terms of size and power. Based on our extensive Monte Carlo simulations, we demonstrate that domination number of our PICD has higher power for certain types of deviations from uniformity compared to other tests.

On some subclasses of interval catch digraphs

“…Then by definition of a i and b i we have b i > a j for all i o ≤ j ≤ i e . 4 Thus b i,j = 1 =⇒ b i > a j .…”

“…Henceforth undirected graphs will be called simply graphs. For some recent applications of ICD and in general, catch digraphs one may consult [4,5,6,13].…”